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April 12th, 2012, 11:01 PM   #1
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Inequalities, number theory

Hello all, I have a question and it may seem a little weird that I should ask it, but I really am confused.

You see, I have just completed a topic on inequalities and was looking at number theory and found that the things I had learned in inequalities did not apply here, or so it seems.
Find the sum of n terms of the GP 3+4.5+6.75+... How large must n be so that this sum is greater than 6000?
I was able to identify common ratio r to be 1.5
Therefore,
Sn>6000
Sn = 3(1-(1.5)^n) / 1-1.5 >6000
3(1-(1.5)^n) / 0.5 > 6000
-6(1-(1.5)^n)>6000
Because I cannot being over a negative with an inequality present,
6((1.5)^n -1) > 6000
((1.5)^n -1) > 1000
And so on till I got n> log1.5(1001)
n> 17.03,
Since I cannot round down,
n>18 ( Answer ) Was my working right? The answer was n>=18, can anyone tell me why?

6((1.5)^n -1) > 6000
((1.5)^n -1) > 1000 This is the part where I'm confused. I vaguely remember that this is how you should approach this problem but what I learned in inequalities told me to in this case, square both sides since LHS and RHS are positive. OHHH actually I just remembered.
I can bring over 6 because it is definitely positive right?
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April 12th, 2012, 11:24 PM   #2
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Re: Inequalities, number theory

I would write:





















You use a weak inequality because n can be 18 or greater. Your working was correct, you just needed a weak inequality at the end.
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April 13th, 2012, 01:23 AM   #3
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Re: Inequalities, number theory

How did you get n-1 and what is a weak inequality? Thank you!
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April 13th, 2012, 05:06 AM   #4
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Re: Inequalities, number theory

I let the summation go from k = 0 to k = n - 1, which is n terms.

A strict inequality is strictly less than or greater than, while a weak inequality is less than or equal to or greater than or equal to.
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